Normalized defining polynomial
\( x^{12} - 2 x^{11} + 11 x^{10} - 8 x^{9} + 34 x^{8} - 18 x^{7} + 23 x^{6} - 12 x^{5} - 2 x^{4} + 14 x^{3} - x^{2} - 4 x + 1 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(207059946098688=2^{12}\cdot 3^{6}\cdot 37^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.60$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{29090} a^{11} + \frac{493}{2909} a^{10} - \frac{4469}{29090} a^{9} - \frac{5441}{29090} a^{8} + \frac{547}{29090} a^{7} + \frac{6961}{29090} a^{6} - \frac{907}{2909} a^{5} - \frac{7377}{29090} a^{4} + \frac{4112}{14545} a^{3} + \frac{4661}{14545} a^{2} - \frac{321}{14545} a + \frac{2231}{14545}$
Class group and class number
$C_{2}$, which has order $2$
Unit group
| Rank: | $5$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{22704}{14545} a^{11} - \frac{7338}{2909} a^{10} + \frac{234464}{14545} a^{9} - \frac{89049}{14545} a^{8} + \frac{724908}{14545} a^{7} - \frac{119786}{14545} a^{6} + \frac{87836}{2909} a^{5} - \frac{74458}{14545} a^{4} - \frac{98284}{14545} a^{3} + \frac{307838}{14545} a^{2} + \frac{85392}{14545} a - \frac{58857}{14545} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 255.355818842 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$\GL(2,Z/4)$ (as 12T52):
| A solvable group of order 96 |
| The 14 conjugacy class representatives for $\GL(2,Z/4)$ |
| Character table for $\GL(2,Z/4)$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.148.1, 6.0.591408.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | 12.4.828239784394752.2, 12.4.828239784394752.1, 12.6.283748815024128.1 |
| Degree 16 siblings: | 16.0.18645702132865499136.1, Deg 16 |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.12.27 | $x^{12} - 18 x^{10} + 171 x^{8} + 116 x^{6} - 313 x^{4} + 190 x^{2} + 877$ | $6$ | $2$ | $12$ | 12T30 | $[4/3, 4/3]_{3}^{4}$ |
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 37 | Data not computed | ||||||